Riemann Hypothesis

THE RIEMANN HYPOTHESISThe Riemann Zeta Function is defined by the interest serial . For s 1 , the series diverges . However , one butt raise that the divergence is not too bad , in the pass with flying colors thatIn position , we have the in suitableities , we find that and so which implies our claimis diminish , as illustrated belowfor s real and 1The situation is more tangled when we imagine the series as a die of a building Byzantine variableis defined by and coincides with the usual turn when s is realIt is not unmanageable to switch off that the complex series is convergent if Re (s 1 . In concomitant , it is absolutely convergent because . cast [2] for the general criteria for convergence of series of functionsInstead , it is a non-trivial task to grow that the Riemann Zeta Function potentiometer be extended far beyond on the complex mat has a pole in s 1It is particularly arouse to evaluate the Zeta Function at detrimental integers . One can prove the following : if k is a arbitrary integer then be defined inductively by : the Bernoulli total with odd index great than 1 are equal to nought . Moreover , the Bernoulli arrives game are all rationalThere is a corresponding formula for the positive integers if n 0 is notwithstanding . The natural question arises : are there any opposite zeros of the Riemann Zeta FunctionRiemann Hypothesis .

Every zero of the Riemann Zeta Function must be either a negative even i nteger or a complex number of real part has ! endlessly many zeros on the precise line Re (s 1Why is the Riemann Zeta function so important in mathematics ? One drive is the strict connection with the scattering of prime add up . For warning , we have a renowned product expansion can be used to prove Dirichlet s theorem on the existence of endlessly many prime numbers in arithmetic progressionfor any s such that Re (s 1 . In fact , we have and it is not difficult to check that this product cannot vanishThe following beautiful picture comes from WikipediaBibliography[1] K . Ireland , M . Rosen , A absolute Introduction to Modern depend Theory , Springer , 2000[2] W . Rudin , Principles of Mathematical Analysis , McGraw pitcher , 1976[3] W . Rudin , Real and multiform Analysis , McGraw Hill , 1986PAGEPAGE 4...If you wishing to get a profuse essay, order it on our website: OrderCustomPaper.com

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